International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.4.1.37
[Vol-4, Issue-1, Jan- 2017] ISSN: 2349-6495(P) | 2456-1908(O)
Regional Flood Frequency Analysis Using Computer Simulations Anusha M1, Surendra H J2 1
Asst. Professor, Dept. of Civil Engineering, Sri Venkateshwara College of Engineering, Bangalore, India 2 Associate Professor, Dept. of Civil Engineering, ATRIA IT, Bangalore, India.
Abstract— Different probability distribution methods were employed to determine the flood frequency analysis using computer simulations. Many probability distributions including Gumbel, lognormal, log-Pearson type iii, General Extreme value have been tried to fit the data. The length of record for most of the stations is over 10 years (chosen from 1956 0nwards). The data was procured from J.R.M.Hosking for various project sites. The common time period of 1956 onwards has been chosen only to avoid the effect of interception of basin due to construction of storage reservoir and was also subject of flood data. The best fitting distribution works out to be General Extreme Value Distribution. Gumbel’s distribution ranks poorly among different probability distributions. A trial version of probability software is used to evaluate the best fit distribution and parameters of distribution. Keywords— Gumbel distribution, General Extreme value, Log normal, Log Pearson distribution. I. INTRODUCTION Natural disaster like Flood is one of the most serious problems that are affecting countries worldwide. Many natural and Un-natural factors affects the flood, monitoring the factors responsible for flood is very important to minimize the impact and damage. Flood real time monitoring system and flood forecasting model is difficult to develop, due to a large amountof data required are difficult to obtain. In the present study, length of record for most of the stations is over 10 years (chosen from 1956 0nwards) and the data was procured from J.R.M Hosking for various project sites. Many papers discuss the estimation of flood frequency using different distribution techniques.[3] used different methods were employed in applying probability distribution in hydrology.[4] develops flood Model to analyze the maximum water level and drainage density using Gumbel distribution. [5] analyzed the frequency ofNyanyadzi River floods using the Gumbel distribution.
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Latest CWC guidelines, spelled out in the manual on estimation of design flood and USGS guidelines are used for analysis.In the present study many distribution techniques such as Gumbel distribution, Gumbel extreme value distribution, Log normal and Log Pearson distribution techniques were adopted to find the frequency of flood occurrence and best method is identified. II. METHODOLOGY The data has been checked for randomness and corrected for high and low outlier. Missing data is generated by correlation of flood peaks. Correction for trend has not been exercised because of absence of long term data for most of the stations. Each catchment is assumed to generate its own peak flood without intervention. Gumbel Extreme value probability distribution by variates was adopted for frequency analysis at the initial stage assuming that most of the stations follow this type of distribution which eventually has proved otherwise. Excel spreadsheet simulation models and probability modeling tools are used for the analysis of the probability statistics. Many probability distributions including Gumbel, lognormal, log-pearson type iii, General Extreme value have been tried to fit the data. The best fitting distribution works out to be General Extreme Value Distribution. Gumbels’s distribution ranks poorly among different probability distributions. A trial version of probability software is used to evaluate the best fit distribution and parameters of distribution. The methods of moments and maximum likelihood are used for evaluation. The parameters obtained from easy fit for distributions like Gumbels (largest EV), lognormal(2p), General extreme value has been considered in working out the flood quintiles for various return periods like 2,2.33,10,25,50,100,200 and 500 years. The different distributions give closer value at small values of return periods and vary considerably at large return periods. The station-wise, distribution-wise graphs depict the variations. Further, distribution- wise graphs for various
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International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.4.1.37 stations also indicates the flood values at different return periods. A) GUMBEL DISTRIBUTION The annual flood peak of a catchment area forms the annual series. The data is arranged in the decreasing order of the magnitude and the probability peak periods of flood being equal or exceeding is estimated by the following equation. P=m/N+1, where, P=probability of exceedance, m= order number. N=total no of events. The recurrence interval T also called return periods of frequency is calculated as, T=1/P Chow has shown that most frequency distributions applicable to hydrologic study can be expressed by the following general equation of hydrologic frequency analysis. Xt= x- +K σ where, Xt = value of the variate of a random hydrologic series in the return period T. x- = the mean of the variate σ = standard deviation of the variate K = frequency factor which depend upon the return period. B) LOG NORMAL DISTRIBUTION Log trans series data to be used. Z=norm lnV(F,0,1) F=1-(1/T) X(f)=e(log mean + log stdev*Z) C) LOG PEARSON DISTRIBUTION F=1-(1/T) Z=Norminv (F,0,1) K=log skew/6 Kt=Z+ (Z2-1)*K+ (Z3-6K)*K2/3-(Z23 4 5 1)*K +Z*K +K /3 X(f)=e(log mean +log stdev(Kt)) D) GENERALIZED EXTREME –VALUE DISTRIBUTION: Parameters (3); ξ (location), α (scale),k (shape) Range of x: -∞ < x ≤ ξ +α/k if k>0; -∞ <x < ∞ if k=0; ξ +α / k ≤ x < ∞ if k<0 f(x) =α-1*e-(1-k)y-e-y ,k≠0 y= {(x- ξ) /α. F(x) =e-e-y
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y={-k-1*log {1-k(x- ξ)/α}
[Vol-4, Issue-1, Jan- 2017] ISSN: 2349-6495(P) | 2456-1908(O)
x (F)={ ξ +α{1-(- log F)k}/ k x (F)={ ξ -α*log(- log F)}
k≠0 k=0
Special cases k=0 is the gumbel distribution, k=1 is a reverse exponential distribution on the interval ξ ≤ x ≤ ξ +α. The three parameters is given by, k= 7.8590c + 2.9554 c2 where, c= (2+ (3+t3))-(log 2 / log 3) α= L2*k / {(1-2-k) β (1+k)} ξ = L1 – α {1- β (1+k)} / k III.
RESULT AND DISCUSSION Table.1: OUTLIER KnVALUES
Sample Size 10
Kn value 2.036
15
2.247
20
2.385
25
2.486
30
2.563
35
2.628
40
2.682
45
2.727
50
2.768
55
2.804
60
2.837
65
2.866
70
2.893
Kn =0.408 ln (n) + 1.158 (for 10 to 100) Kn =0.352 ln (n) + 1.394 (for 60 to 100) Kn =0.443 ln (n) + 1.040 (for 10 to 50) Kn =0.364 ln (n) + 1.342 (for 50 to 100) Rule a) High outlier must be greater than the maximum value. b) Low outlier must be lesser than the minimum value. Since it is difficult to show all the 44 stations, hence flood value for different return periods is shown only for four stations. We can also compare the flood values for different distributions.
k=0
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International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.4.1.37 Table.2: FLOOD VALUE BY GUMBEL 2 1321 1380 1640 1851 2118 2316 2512 2708 2966
[Vol-4, Issue-1, Jan- 2017] ISSN: 2349-6495(P) | 2456-1908(O)
T \ STATIONS 2 2.33 5 10 25 50 100 200 500
1 1537 1570 1712 1827 1973 2081 2189 2296 2437
3 1431 1552 2076 2503 3043 3443 3840 4236 4758
4 1350 1409 1668 1879 2146 2343 2540 2735 2993
T \ STATIONS 2 2.33 5 10 25 50 100 200 500
Table.3: FLOOD VALUE BY LOGNORMAL 1 2 3 1558 1337 1425 1594 1399 1535 1733 1655 2027 1832 1850 2438 1944 2084 2967 2020 2251 3369 2090 2412 3777 2157 2569 4193 2189 2645 4339
4 1366 1428 1684 1879 2111 2276 2435 2591 2666
T \ STATIONS 2 2.33 5 10 25 50 100 200 500
Table.4: FLOOD VALUE BY LOG PEARSON 1 2 3 1560 1324 1322 1596 1385 1427 1734 1651 1982 1831 1863 2574 1939 2128 3554 2012 2324 4496 2079 2520 5665 2142 2717 7121 2221 2982 9623
4 1299 1359 1658 1954 2412 2827 3320 3910 4882
T \ STATIONS 2
Table.5: FLOOD VALUE BY GUMBEL(EXTREME VALUE) DISTRIBUTION 1 2 3 1563 1314 1299
4 1350
2.33
1601
1377
1394
1410
5
1746
1651
1894
1659
10 25 50 100
1842 1943 2005 2056
1876 2165 2383 2600
2427 3315 4174 5245
1850 2076 2233 2381
200
2100
2820
6582
2521
500
2148
3113
8878
2694
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International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.4.1.37
[Vol-4, Issue-1, Jan- 2017] ISSN: 2349-6495(P) | 2456-1908(O)
T v/s Qmax
y = 161.94ln(x) + 1441.6 R² = 0.9989
3500.0 3000.0
Q max
2500.0 2000.0 1500.0 1000.0
500.0 0.0 0
50
100
150
200
250
300
350
400
450
500
T in years Fig.1: COMPARISON OF T v/s Qmax FOR GUMBEL AND LOGNORMAL DISTRIBUTION
CA v/s MAF for the region
2000
Q = 34.271A0.8321 R² = 0.7925
1800
MAF IN CUMECS
1600 1400
1200 1000 800 600 400 200 0 0.000
20.000
40.000 60.000 80.000 CATCHMENT AREA IN SQ.KM
100.000
120.000
Fig.2: Regional Flood Equation arrived for the region IV. CONCLUSIONS a) Gumbel distribution does not too close as compare to lognormal and log Pearson distributions. Only GEV and LN3 distributions fit well for most of the sites. b) Gumbel distribution used for modeling gives different magnitudes of flood quantiles at high return periods. c) The regional flood equation arrived at for region is
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Q=34.271A0.8321 with R2=0.7925 d) Accuracy of the predicted flood values depends primarily on the accuracy of the data. This study being based on annual maximum stream flow data available in Hosking book. e) This flood values obtained will be different for different methods. It is based upon the terrain, land Page | 234
International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.4.1.37
f)
cover and land use pattern. With the available data these methods will follow their own pattern to determine the flood value. Discordancy test is conducted for all the stations, all stations are having the value less than three except one station which is having a value more than three, which will be discorded.
[Vol-4, Issue-1, Jan- 2017] ISSN: 2349-6495(P) | 2456-1908(O)
[7] Satish Chandra, Director, Regional Flood Frequency Analysis, National Institute of Hydrology, Jal Vigyan Bhavan, Roorkee. [8] Subramanya.K,(2006) “Engineering Hydrology” ii Edition, Tata McGraw Hill Publishing Company Ltd, New Delhi. [9] Ven Te Chow “Applied hydrology McGraw hill Editions, New Delhi.
ACKNOWLEDGEMENT Authors are grateful to Sri C.S.NAGENDRA Research officer and Sri. K.R.SATHYANARAYANA Assistant research officer, K.E.R.S, Krishnaraja Sagara, Government of Karnataka, for their valuable guidance encouragement and co-operation in all stages of work. ABBREVIATIONS CA: Catchment area, GEV: Generalized extreme value, m:order number, ξ:Location parameter,N:Number of observation, P or F:Probability of exceedance, PDF:Probability Density Function,2P:Two parameters,Q:Flood peak,R:Regression coefficient, σ: Standard deviation, T: Return period, USGS: United State Geological Survey, k: Shape parameters. REFERENCES [1] Developing a methodology for estimation of design flood in southern region of Karnataka, modified report research scheme applied to flood control project, sponsored by CBIP KERS, K.R.SAGAR. [2] D van Bladeren.,P K Zawada and D Mahlangu. “Statistical Based Regional Flood Frequency Estimation Study for South Africa Using Systematic, Historical and Palaeoflood Data, Pilot Study – Catchment Management Area 15”. WRC Report No 1260/1/07, ISBN 078-1-77005-537-7,March 2007. [3] Gh.Khosravi.,A.Majidi., A.Nohegar.” Determination of suitability probability distribution for annual mean and peak discharges Estimation ( case study : Minab river-Barantin Gage, Iran). International Journal of Probability and statistics, 2012, 1(5): 160-163. [4] Kantima M., Pattorawait P. “Mathematical modeling for flood forecasting in the chi river: Amphur Muang, Khon Kaen, Thailand case”. International conference on computer science and Information Technology, December 2011. [5] Never Mujere. “Flood frequency analysis by Gumbel Distribution”. International Journal on computer science and Engineering, vol:3, no-7, july-2011. [6] Regional frequency analysis by HOSKING and WALLIS. www.ijaers.com
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